Properties

Label 97.49
Modulus $97$
Conductor $97$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(97, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([31]))
 
pari: [g,chi] = znchar(Mod(49,97))
 

Basic properties

Modulus: \(97\)
Conductor: \(97\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 97.k

\(\chi_{97}(2,\cdot)\) \(\chi_{97}(3,\cdot)\) \(\chi_{97}(11,\cdot)\) \(\chi_{97}(25,\cdot)\) \(\chi_{97}(31,\cdot)\) \(\chi_{97}(32,\cdot)\) \(\chi_{97}(44,\cdot)\) \(\chi_{97}(48,\cdot)\) \(\chi_{97}(49,\cdot)\) \(\chi_{97}(53,\cdot)\) \(\chi_{97}(65,\cdot)\) \(\chi_{97}(66,\cdot)\) \(\chi_{97}(72,\cdot)\) \(\chi_{97}(86,\cdot)\) \(\chi_{97}(94,\cdot)\) \(\chi_{97}(95,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\(5\) → \(e\left(\frac{31}{48}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 97 }(49, a) \) \(1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{13}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 97 }(49,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(49,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 97 }(49,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2